Global attractivity of a unique positive periodic solution for a first-order nonlinear difference equation with time delays

The present paper is directed toward the study on the global attractivity of a unique positive periodic solution of a discrete hematopoiesis model with unimodal production functions and several time delays. This model is described by a nonlinear difference equation. The result obtained is proved by transforming this model into another difference equation and by using the Schauder fixed-point theorem.     

Positive solution for nonlinear third-order multi-point boundary value problem at resonance

In this paper, positive solutions for a kind of third-order multi-point boundary value problem at resonance are investigated. By using the Leggett-Williams norm-type theorem due to O''Regan and Zima, existence result of at least one positive solution is established. An example is given to demonstrate the main results.     

Total positivity of Narayana matrices

We prove the total positivity of the Narayana triangles of type $A$ and type $B$, and thus affirmatively confirm a conjecture of Chen, Liang and Wang and a conjecture of Pan and Zeng. We also prove the strict total positivity of the Narayana squares of type $A$ and type $B$.     

Banach 空间中一阶周期边值问题的广义解

In this paper, the authors study the periodic boundary value problems of a class of nonlinear integro-differential equations of mixed type in Banach space with Carath′eodory's conditions. We arrive at the conclusion of the existence of generalized solutions between general- ized upper and lower solutions, and develop the monotone iterative technique to find generalized extremal solutions as limits of monotone solution sequences in Banach space.     

Interior proximal methods and central paths for convex second-order cone programming

We make a unified analysis of interior proximal methods of solving convex second-order cone programming problems. These methods use a proximal distance with respect to second-order cones which can be produced with an appropriate closed proper univariate function in three ways. Under some mild conditions, the sequence generated is bounded with each limit point being a solution, and global rates of convergence estimates are obtained in terms of objective values. A class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution. These results are illustrated with some examples. In addition, we also study the central paths associated with these distance-like functions, and for the linear SOCP we discuss their relations with the sequence generated by the interior proximal methods. From this, we obtain improved convergence results for the sequence for the interior proximal methods using a proximal distance continuous at the boundary of second-order cones.     

Tutorial on evaluation of type I and type II errors in chemical analyses: From the analytical detection to authentication of products and process control

Uncertainty is inherent in all experimental determinations. Nevertheless, these measurements are used to make decisions including the performance of the own measurement systems. The link between the decision and the true implicit system that generates the data (measurement system, production process, category of samples, etc.) is a representation of this uncertainty as a probability distribution. This representation leads to the probabilistic formalization of the possibility of making errors. In the context of regulations established by official agencies, it is important to use these statistical decision methods in some cases because the own norm makes them mandatory and, in general, because this is the way of reasonably evaluating whether a working hypothesis is rejected on the basis of the experimental data.The aim of the present tutorial is to introduce some ideas and basic methods for the critical analysis of experimental data. With this goal, the basic elements of the Neyman-Pearson theory of hypothesis testing are formally introduced in connection with the common problems in chemical analysis and, if this is the case, their relation to the norms of regulatory agencies. The notion of decision with ‘enough quality’ is modelled when explicitly considering: (1) the null, H₀, and alternative, H₁, hypotheses. (2) The significance level of the test, which is the probability, α, of rejecting H₀ when it is true, and the power of the test, 1 − β, β being the probability of accepting H₀ when it is false. (3) The difference between H₀ and H₁ that has to be detected with experimental data. (4) The needed sample size. These four concepts should be explicitly defined for each problem and, under the usual assumption of normal distribution of the data, the mathematical relations among these concepts are shown, which allow the analyst to design a decision rule with pre-set values of α and β.To illustrate the unifying character of this inferential methodology, several situations are exposed along the tutorial: the design of a hypothesis test to decide on the performance characteristics of analytical methods, the capability of detection of both quantitative and qualitative analytical methods (including its generalization to the case of multivariate and/or multiway signals), the analytical sensitivity with multivariate signals, the class-modelling and the process control.     

Generation of gas‐phase sodiated arenes such as [(Na3(C6H4)+] from benzene dicarboxylate salts

Upon collision‐induced activation, gaseous sodium adducts generated by electrospray ionization of disodium salts of 1,2‐ 1,3‐, and 1,4‐benzene dicarboxylic acids (m/z 233) undergo an unprecedented expulsion of CO₂ by a rearrangement process to produce an ion of m/z 189 in which all three sodium atoms are retained. When isolated in a collision cell of a tandem‐in‐space mass spectrometer, and subjected to collision‐induced dissociation (CID), only the m/z 189 ions derived from the meta and para isomers underwent a further CO₂ loss to produce a peak at m/z 145 for a sodiated arene of formula (Na₃C₆H₄)⁺. This previously unreported m/z 145 ion, which is useful to differentiate meta and para benzene dicarboxylates from their ortho isomer, is in fact the sodium adduct of phenelenedisodium. Moreover, the m/z 189 ion from all three isomers readily expelled a sodium radical to produce a peak at m/z 166 for a radical cation [(^?C₆H₄CO₂Na₂)⁺], which then eliminated CO₂ to produce a peak at m/z 122 for the distonic cation (^?C₆H₄Na₂)⁺. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with p‐Laplacian in Banach space

This paper deals with 2 core aspects of fractional calculus including existence of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with nonlinear p‐Laplacian operator in Caputo sense. For these aims, the suggested problem is converted into an integral equation via Green function , for ε∈(n−1,n], where n≥4. Then, the Green function is examined whether it is increasing or decreasing and positive or negative function. After these properties, some classical fixed‐point theorems are used for the existence of positive solution. Hyers‐Ulam stability of the proposed problem is also considered. For the application of the results, an expressive example is included.